Abstract

Stimulated emission is formulated in completely classical terms and is shown to occur in general only in nonlinear systems. Our approach is based on the frequency-dependent susceptibility, which in both the classical and quantum-mechanical descriptions is the main characteristic determining whether there is absorption or stimulated emission. By using Bom’s correspondence rule, we derive an expression for the lowest-order quantum correction to the classical susceptibility.

© 1987 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Einstein, Physik. Z. 18, 21 (1917).
  2. A. Gaponov, Sov. Phys. JETP 12, 282 (1961); A. Gaponov, M. Petelin, and V. Yulpatov, Radiophysics 10, 1414 (1967) (in Russian).
  3. B. Fain, Sov. Phys. JETP 23, 882 (1966); B. Fain and Y. I. Khanin, Quantum Electronics (MIT Press, Cambridge, Mass., 1966), Vol. I; Photons and Nonlinear Media (Nauka, Moscow, 1972) (in Russian).
  4. M. Cray, M.-L. Shih, and P. W. Milonni, Am. J. Phys. 50, 1016 (1982).
    [Crossref]
  5. L. Landau and E. Lifshitz, Quantum Mechanics (Pergamon, New York, 1977).
  6. L. Landau and E. Lifshitz, Mechanics (Pergamon, New York, 1966).
  7. M. Born, Z. Phys. 26, 379 (1924). See also the discussion by M. Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966), p. 193.
    [Crossref]
  8. I. Sobelman and I. Tutin, Usp. Fiz. Nauk 79, 595 (1963) (Russian).
  9. H. A. Lorentz, The Theory of Electrons (Dover, New York, 1952).
  10. See, for instance, the review by P. W. Milonni, Phys. Rep. 25, 1 (1976); Am. J. Phys. 52, 340 (1984). The physical interpretation of spontaneous emission is also discussed by B. Fain, Nuovo Cimento B 68, 73 (1982).
    [Crossref]
  11. This point is discussed by P. W. Milonni, Am. J. Phys. 49, 177 (1981).
    [Crossref]

1982 (1)

M. Cray, M.-L. Shih, and P. W. Milonni, Am. J. Phys. 50, 1016 (1982).
[Crossref]

1981 (1)

This point is discussed by P. W. Milonni, Am. J. Phys. 49, 177 (1981).
[Crossref]

1976 (1)

See, for instance, the review by P. W. Milonni, Phys. Rep. 25, 1 (1976); Am. J. Phys. 52, 340 (1984). The physical interpretation of spontaneous emission is also discussed by B. Fain, Nuovo Cimento B 68, 73 (1982).
[Crossref]

1966 (1)

B. Fain, Sov. Phys. JETP 23, 882 (1966); B. Fain and Y. I. Khanin, Quantum Electronics (MIT Press, Cambridge, Mass., 1966), Vol. I; Photons and Nonlinear Media (Nauka, Moscow, 1972) (in Russian).

1963 (1)

I. Sobelman and I. Tutin, Usp. Fiz. Nauk 79, 595 (1963) (Russian).

1961 (1)

A. Gaponov, Sov. Phys. JETP 12, 282 (1961); A. Gaponov, M. Petelin, and V. Yulpatov, Radiophysics 10, 1414 (1967) (in Russian).

1924 (1)

M. Born, Z. Phys. 26, 379 (1924). See also the discussion by M. Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966), p. 193.
[Crossref]

1917 (1)

A. Einstein, Physik. Z. 18, 21 (1917).

Born, M.

M. Born, Z. Phys. 26, 379 (1924). See also the discussion by M. Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966), p. 193.
[Crossref]

Cray, M.

M. Cray, M.-L. Shih, and P. W. Milonni, Am. J. Phys. 50, 1016 (1982).
[Crossref]

Einstein, A.

A. Einstein, Physik. Z. 18, 21 (1917).

Fain, B.

B. Fain, Sov. Phys. JETP 23, 882 (1966); B. Fain and Y. I. Khanin, Quantum Electronics (MIT Press, Cambridge, Mass., 1966), Vol. I; Photons and Nonlinear Media (Nauka, Moscow, 1972) (in Russian).

Gaponov, A.

A. Gaponov, Sov. Phys. JETP 12, 282 (1961); A. Gaponov, M. Petelin, and V. Yulpatov, Radiophysics 10, 1414 (1967) (in Russian).

Landau, L.

L. Landau and E. Lifshitz, Quantum Mechanics (Pergamon, New York, 1977).

L. Landau and E. Lifshitz, Mechanics (Pergamon, New York, 1966).

Lifshitz, E.

L. Landau and E. Lifshitz, Mechanics (Pergamon, New York, 1966).

L. Landau and E. Lifshitz, Quantum Mechanics (Pergamon, New York, 1977).

Lorentz, H. A.

H. A. Lorentz, The Theory of Electrons (Dover, New York, 1952).

Milonni, P. W.

M. Cray, M.-L. Shih, and P. W. Milonni, Am. J. Phys. 50, 1016 (1982).
[Crossref]

This point is discussed by P. W. Milonni, Am. J. Phys. 49, 177 (1981).
[Crossref]

See, for instance, the review by P. W. Milonni, Phys. Rep. 25, 1 (1976); Am. J. Phys. 52, 340 (1984). The physical interpretation of spontaneous emission is also discussed by B. Fain, Nuovo Cimento B 68, 73 (1982).
[Crossref]

Shih, M.-L.

M. Cray, M.-L. Shih, and P. W. Milonni, Am. J. Phys. 50, 1016 (1982).
[Crossref]

Sobelman, I.

I. Sobelman and I. Tutin, Usp. Fiz. Nauk 79, 595 (1963) (Russian).

Tutin, I.

I. Sobelman and I. Tutin, Usp. Fiz. Nauk 79, 595 (1963) (Russian).

Am. J. Phys. (2)

M. Cray, M.-L. Shih, and P. W. Milonni, Am. J. Phys. 50, 1016 (1982).
[Crossref]

This point is discussed by P. W. Milonni, Am. J. Phys. 49, 177 (1981).
[Crossref]

Phys. Rep. (1)

See, for instance, the review by P. W. Milonni, Phys. Rep. 25, 1 (1976); Am. J. Phys. 52, 340 (1984). The physical interpretation of spontaneous emission is also discussed by B. Fain, Nuovo Cimento B 68, 73 (1982).
[Crossref]

Physik. Z. (1)

A. Einstein, Physik. Z. 18, 21 (1917).

Sov. Phys. JETP (2)

A. Gaponov, Sov. Phys. JETP 12, 282 (1961); A. Gaponov, M. Petelin, and V. Yulpatov, Radiophysics 10, 1414 (1967) (in Russian).

B. Fain, Sov. Phys. JETP 23, 882 (1966); B. Fain and Y. I. Khanin, Quantum Electronics (MIT Press, Cambridge, Mass., 1966), Vol. I; Photons and Nonlinear Media (Nauka, Moscow, 1972) (in Russian).

Usp. Fiz. Nauk (1)

I. Sobelman and I. Tutin, Usp. Fiz. Nauk 79, 595 (1963) (Russian).

Z. Phys. (1)

M. Born, Z. Phys. 26, 379 (1924). See also the discussion by M. Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966), p. 193.
[Crossref]

Other (3)

H. A. Lorentz, The Theory of Electrons (Dover, New York, 1952).

L. Landau and E. Lifshitz, Quantum Mechanics (Pergamon, New York, 1977).

L. Landau and E. Lifshitz, Mechanics (Pergamon, New York, 1966).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (2)

Fig. 1
Fig. 1

Energy levels of an anharmonic oscillator (see the text).

Fig. 2
Fig. 2

Plot of the function Y = −ν4[(ωωk)2ν2]/[(ωωk)2 + ν2]3 versus X = (ωωk)/ν.

Equations (104)

Equations on this page are rendered with MathJax. Learn more.

H ˆ = H ˆ 0 ( x ˆ , p ˆ ) e x ˆ E ( t ) .
d H ˆ 0 d t = ( i / ћ ) [ H ˆ , H ˆ 0 ] = ( i e / ћ ) [ H ˆ 0 , x ˆ ] E ( t ) = e x ˙ ˆ E ( t )
d H 0 d t = [ H , H 0 ] = e x ˙ E ( t )
x ˙ = x ˙ 0 + x ˙ 1 ,
e x ˙ 1 = d d t [ χ ( ω ) E ( ω ) e i ω t + χ ( ω ) E ( ω ) e i ω t ] .
x ( t ) = x ( ω ) e i ω t + x ( ω ) e i ω t ,
e x ( ω ) = χ ( ω ) E ( ω ) .
x 0 ( t ) = x 0 e i ω 0 t + x 0 * e i ω 0 t ,
d H 0 d t = e x ˙ 0 E ( t ) = 2 e ω 0 | x 0 E ( ω ) | sin [ ( ω ω 0 ) t + ψ φ ] .
d H 0 d t = 2 ω χ ( ω ) | E ( ω ) | 2 ,
H ˆ = H ˆ X + H ˆ Y a x ˆ a y ˆ a ,
x ˆ a ( t ) = 0 d ω [ A ˆ a ( ω ) e i ω t + A ˆ a ( ω ) e i ω t ] .
Δ H X i = i 0 d ω ω [ χ a b ( ω ) χ ab * ( ω ) ] A ˆ a ( ω ) A ˆ b ( ω ) ,
ρ ˆ ( t ) ρ ˆ ( 0 ) + i ћ 0 t d t 1 [ x ˆ a ( t 1 ) y ˆ a ( t 1 ) , ρ ˆ ( 0 ) ] .
χ a b ( ω ) = i ћ 0 d τ [ y ˆ a ( τ ) , y ˆ b ( 0 ) ] exp [ i ( ω + i ) t ] , 0 + .
Δ H X s = 0 d ω { 2 π ћ ( y ˆ b y ˆ a ) ω + i ω 2 [ χ a b ( ω ) χ ba * ( ω ) ] } × A ˆ b ( ω ) , A ˆ a ( ω ) ,
( y ˆ a y ˆ b ) ω ( 1 / 4 π ) d τ y ˆ a ( 0 ) y ˆ b ( τ ) + y ˆ b ( τ ) y ˆ a ( 0 ) e i ω τ .
A ˆ a ( t ) = 0 d ω A ˆ a ( ω ) e i ω t ,
A ˆ a ( t ) = 0 d ω A ˆ a ( ω ) e i ω t ,
A ˆ a ( t ) A ˆ b ( t 1 ) = 0 d ω 0 d ω A ˆ a ( t ) A ˆ b ( t 1 ) × exp [ i ( ω t ω t 1 ) ] .
A ˆ a ( ω ) A ˆ b ( ω ) = ( 1 / 2 π 2 ) d t d t 1 A ˆ a ( t ) A ˆ b ( t 1 ) × exp [ i ( ω t ω t 1 ) ] ,
Δ H X i = ( 1 / 2 π ) d t d τ S a b ( τ ) A ˆ a ( t ) A ˆ b ( t τ ) ,
S a b ( τ ) = i 0 d ω ω [ χ a b ( ω ) χ b a * ( ω ) ] e i ω τ .
ω 0 δ ω Δ ω ,
S a b ( τ ) i ω 0 [ χ a b ( ω 0 ) χ b a * ( ω 0 ) ] d ω e i ω τ = 2 π i δ ( τ ) ω 0 [ χ a b ( ω 0 ) χ b a * ( ω 0 ) ] .
Δ H X i d t | i ω 0 [ χ a b ( ω 0 ) χ b a * ( ω 0 ) ] A ˆ a ( t ) A ˆ b ( t ) | .
H ˙ X i i ω 0 [ χ a b ( ω 0 ) χ b a * ( ω 0 ) ] A ˆ a ( t ) A ˆ b ( t ) ,
d ˆ ( t ) = k = k = d ˆ k ( E ) e i k φ = k = k = d ˆ k ( t ) .
φ = ω 0 t + φ 0 .
χ ( ω ) = i ћ k n 0 [ d ˆ k ( τ ) , d ˆ n ( 0 ) ] exp [ i ( ω + i ) τ ] d τ .
ω ω k = k ω 0 ( E )
χ ( ω ) = i ћ 0 [ d ˆ k ( τ ) , d ˆ k ( 0 ) ] exp i ( ω + i ) τ ] d τ
[ A ˆ , B ˆ ] = i ћ n ( δ A ˆ δ q n δ B ˆ δ p n δ A ˆ δ p n δ B ˆ δ q n ) .
q ˆ n = q ˆ n + δ q ˆ n q n + δ q n ,
p ˆ n = p ˆ n + δ p ˆ n p n + δ p ˆ n .
A ˆ = A + n ( δ A δ q n δ q ˆ n + δ A δ p n δ p ˆ n ) + ,
B ˆ = B + n ( δ B δ q n δ q ˆ n + δ B δ p n δ p ˆ n ) +
[ A ˆ , B ˆ ] i ћ n ( δ A δ q n δ B δ p n δ A δ p n δ B δ q n ) ,
δ E δ I = ω 0 ( E ) ,
[ d ˆ k ( τ ) , d ˆ k ( 0 ) ] ћ ω k δ δ E [ d k ( τ ) d k ( 0 ) ]
χ ( ω ) = f ( E ) d E 0 i ω k δ δ E [ d k ( τ ) d k ( 0 ) ] exp i ( ω + i ) τ ] d τ ,
f ( E ) d E = 1
d k ( τ ) = d k exp ( i ω k ν ) τ ,
χ ( ω ) = f ( E ) ω k ( E ) δ δ E ( | d k | 2 ω ω k ( E ) + i ν ( E ) ) d E .
E = 1 2 m ω 0 2 x 2 + 1 2 m x ˙ 2
| d ± | 2 = e 2 E 2 m ω 0 2 .
χ ( ω ) = e 2 2 m ω 0 1 ω ω 0 + i ν ,
χ ( ω ) = e 2 2 m ω 0 1 ( ω ω 0 ) 2 + ν 2 .
χ ( ω ) = δ δ E [ f ( E ) ω k ] | d k | 2 ν ( ω ω k ) 2 + ν 2 d E .
ω = ω k ( E )
E = E 0 ,
Δ ω k ( E ) = | δ ω k ( E ) / δ E | E = E 0 Δ E > ν ,
δ δ E [ f ( E ) ω k ( E ) ] > 0 ,
Δ E = ћ ω 0 ( E ) Δ n ,
f ( E ) Δ E = f ( E ) ћ ω 0 Δ n = n n + Δ n ρ m m ρ n n Δ n ,
f ( E ) ћ ω 0 = ρ n n .
χ ( ω ) = n ω n ( E 0 ) δ δ E [ | d n | 2 ν ( ω ω n ) 2 + ν 2 ] | E = E 0 .
χ ( ω ) = ω n δ δ E ( | d n | 2 ν ) ( ω ω n ) 2 + ν 2 2 ω n ν 2 δ ν δ E | d n | 2 [ ( ω ω n ) 2 + ν 2 ] + 2 ω n | d n | 2 ν ( ω ω n ) δ ω n δ E [ ( ω ω n ) 2 + ν 2 ] 2 .
( ω ω n ) δ ω n δ E < 0.
ω > ω n = n ω 0 , δ ω n / δ E < 0 ,
Δ H = 4 π 2 k d E | d k | 2 ω k | A ( ω k ) | 2 δ δ E [ ω k f ( E ) ] = 4 π 2 k d E ω k f ( E ) δ δ E [ | A ( ω k ) | 2 | d k | 2 ω k ] = 4 π 2 k ω k δ δ E [ | A ( ω k ) | 2 | d k | 2 ω k ] ,
Δ H = 4 π 2 | A | 2 k ω k δ δ E [ ω k | d k | 2 ] .
f n n = ( 2 m / ћ ) ω n n | x n n | 2 .
n f n n = n > 0 ( | f n , n 1 | | f n , n + 1 | ) = 1.
k ω k δ δ E [ ω k | d k | 2 ] > 0 ,
V ( x , t ) = k = k = V k ( E , ω ) e i ω t e i k φ d ω .
Δ H = 4 π 2 k ω k δ δ E [ ω k V k ( E , ω k ) ] .
[ A ˆ , B ˆ ] = i ћ [ A , B ] + i ћ 2 [ ( δ 2 A δ q 2 δ 2 B δ 2 p δ 2 A δ p 2 δ 2 B δ q 2 ) × δ q ˆ δ p ˆ + δ p ˆ δ q ˆ + ( δ 2 A δ q 2 δ 2 B δ q δ p δ 2 B δ q 2 δ 2 A δ q δ p ) × δ q ˆ 2 ( δ 2 A δ p 2 δ 2 B δ q δ p δ 2 B δ p 2 δ 2 A δ q δ p ) δ p ˆ 2 ]
k δ A δ n A ( n ) A ( n k ) ,
k δ A ( n ) δ n k δ A δ n 1 2 k 2 δ 2 A δ n 2 + .
δ A δ n = δ A δ E δ E δ n = ћ ω 0 ( E ) δ A δ E ,
δ 2 A δ n 2 = ћ 2 ω 0 δ δ E ( ω 0 δ A δ E ) .
k δ A δ n ћ ω k δ A δ E 1 2 ћ 2 ω k δ δ E ( ω k δ A δ E ) ,
ω k δ δ E [ d k ( τ ) d k ( 0 ) ] = k ω 0 δ n δ E δ δ n [ d k ( τ ) d k ( 0 ) ] = 1 ћ k δ δ n [ d k ( τ ) d k ( 0 ) ] ω k δ δ E [ d k ( τ ) d k ( 0 ) ] 1 2 ћ ω k δ δ E × { ω k δ δ E [ d k ( τ ) d k ( 0 ) ] } + .
χ ( ω ) = f ( E ) d E 0 i ω k δ δ E [ d k ( τ ) d k ( 0 ) ] exp [ i ( ω + i ) ] τ ] d τ ( i ћ / 2 ) f ( E ) d E 0 ω k δ δ E { ω k δ δ E [ d k ( τ ) d k ( 0 ) ] } × exp [ i ( ω + i ) τ ] d τ ,
χ q ( ω ) = ћ 2 d E f ( E ) ω k δ δ E [ ω k δ δ E | d k | 2 ω ω k ( E ) + i ν ] .
χ q ( ω ) = ( ћ / 2 ) d E f ( E ) ω k δ δ E { ω k δ δ E | d k | 2 [ ω ω k ( E ) ] 2 + ν 2 } .
χ q ( ω ) = ( ћ ω k / 2 ) δ δ E χ cl ( ω ) ,
χ ( ω ) = χ q ( ω ) + χ cl ( ω ) = χ cl ( ω ) ( ћ ω k / 2 ) δ δ E χ cl ( ω ) .
δ L ( ω ω k ) = 1 π ν [ ω ω k ( E ) ] 2 + ν 2 ,
χ cl ( ω ) = π ω k [ δ L ( ω ω k ) δ δ E | d k | 2 | d k | 2 δ L ( ω ω k ) δ ω k δ E ]
χ q ( ω ) = π ћ ω k 2 δ δ E [ ω k δ L ( ω ω k ) δ δ E | d k | 2 ω k | d k | 2 δ L ( ω ω k ) δ ω k δ E ] = π ћ ω k 2 [ δ δ E ω k δ δ E ( | d k | 2 ) δ L ( ω ω k ) ω k δ δ E ( | d k | 2 ) δ L ( ω ω k ) δ ω k δ E δ δ E ( | d k | 2 δ ω k δ E ) ω k δ L ( ω ω k ) + ω k | d k | 2 δ L ( ω ω k ) ( δ ω k δ E ) 2 ] ,
δ δ E δ L ( ω ω k ) δ L ( ω ω k ) δ ω k δ E .
Δ χ q ( ω ) = π 2 ћ ω k 2 | d k | 2 ( δ ω k δ E ) 2 δ L ( ω ω k ) = π ћ ω k 2 2 | d k | 2 ( δ ω k δ E ) 2 2 ν π ( ω ω k ) 2 ν 2 [ ( ω ω k ) 2 + ν 2 ] 3 .
ω = ω k ± 2 ν .
Δ χ q ( ω k ± 2 ν ) < 0.
R = Δ χ q ( ω = ω k , ω k ± 2 ν ) χ cl ( ω = ω k ) ћ ω k | d k | 2 δ δ E | d k | 2 ( δ ω k δ E ) 2 ν 2 Δ ω k ω k ν 2 ,
Δ ω k = ћ ω k δ ω k δ E
R Δ ω k ω l ( ω k ν ) 2 .
Δ ω k / ω k = ( ћ ω k / ω k ) δ ω k δ E = ћ ω k ( ω k / δ ω k δ E ) 1 = ћ ω k / E * ,
E ( n ) = I 0 / n 2 ,
ω 0 ( n ) = ( 1 / ћ ) δ E ( n ) δ n = 2 I 0 / ( ћ n 3 ) ,
ω 0 ( E ) = ( 1 / ћ ) ( E 3 / I 0 ) 1 / 2 ,
δ ω 0 δ E = ( 3 / 2 ћ ) ( E / I 0 ) 1 / 2 ,
E * = 2 E / 3 .
Δ ω 0 / ω 0 = 3 ћ ω 0 / 2 E = ( 3 / 2 ) ( E / I 0 ) 1 / 2 .
Δ ω 0 / ω 0 = ( 3 ћ / 2 ) ( 2 E / μ e 4 ) .
U ( x ) = D ( e 2 a x 2 e a x ) ,
E ( n ) = ( ћ Ω / 4 σ ) [ 2 σ ( n + 1 / 2 ) ] 2 ,
σ = D / ћ Ω and Ω = a ( D / μ ) 1 / 2 ,
ћ ω 0 = δ E ( n ) δ n = ћ Ω ( 1 n + 1 / 2 2 σ ) = 2 ( ћ Ω E / σ ) 1 / 2
ω 0 = 2 Ω ( E / D ) 1 / 2 and δ ω 0 δ E = Ω ( D E ) 1 / 2 .
E * = ω 0 ( δ ω 0 δ E ) 1 = 2 E
Δ ω 0 / ω 0 = ћ Ω ( E D ) 1 / 2 = ћ a ( μ E ) 1 / 2 .

Metrics